If you drew rays from the center of the earth out to infinity at every angle, what percentage of them would intersect a star?

Extra details:

Assume the rays are mathematical rays, or that they travel at infinite speeds.

Even in an infinite universe, since the apparent magnitude of stars decreases with distance, it is conceivable that the area occluded is less than 1, i.e. that the total magnitude of the stars at a given distance approaches zero as the distance increases.

Your question is a classical paradox (why the skynight is not bright) known as Olber's paradox. I don't have the exact percentage, but you should edit your question ask it with or without space expansion.
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ShaktyaiAug 24 '12 at 17:11

Thank you! But could you clarify why space expansion makes a difference if the thought experiment happens at a single instant?
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Matthew PiziakAug 24 '12 at 17:17

"Suppose that the universe were not expanding, and always had the same stellar density; then the temperature of the universe would continually increase as the stars put out more radiation. Eventually, it would reach 3000K (corresponding to a typical photon energy of 0.3 eV and so a frequency of 7.5×1013 Hz), and the photons would begin to be absorbed by the hydrogen plasma filling most of the universe, rendering outer space opaque. .... So the sky is about fifty billion times darker than it would be if the universe were neither expanding nor too young to have reached equilibrium yet."
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ShaktyaiAug 24 '12 at 17:32

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It's written in the link I have provided
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ShaktyaiAug 24 '12 at 17:32

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-1: why are you repeating an old paradox? I doubt that you do not already know the answer. The answer is the ratio of starlight to sunlight times the fraction of the sun's disk in the sky. The reason it isn't infinite is because most rays hit the cosmological horizon, or rather, end on the source of microwave background light, which is space-filling.
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Ron MaimonAug 24 '12 at 20:10

1 Answer
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With the assumptions you have added to the question the answer would be that every ray would end on a star and that the night sky would be as bright as the surface of the average star - i.e. quite bright.

You are essentially describing Obler's paradox. Per Wikipedia:

In astrophysics and physical cosmology, Olbers' paradox, named after
the German astronomer Heinrich Wilhelm Olbers and also called the
"dark night sky paradox", is the argument that the darkness of the
night sky conflicts with the assumption of an infinite and eternal
static universe. The darkness of the night sky is one of the pieces of
evidence for a non-static universe such as the Big Bang model. If the
universe is static and populated by an infinite number of stars, any
sight line from Earth must end at the (very bright) surface of a star,
so the night sky should be completely bright. This contradicts the
observed darkness of the night. (see http://en.wikipedia.org/wiki/Olbers%27_paradox)

With the questions assumptions of a "snapshot" of the universe and that light travels at infinite speed along with the additional assumption that the universe is infinite,the result would be a very bright sky. Now the measured curvature of the universe is flat to within about 1%, so if the universe is in fact flat or even negatively curved the universe would be infinite and you would obtain Oblers' paradox. If it turns out that the universe is positively curved it would be finite but you would still have Oblers' paradox since the ray that "circles" the universe without hitting a star would keep circling the universe until it does hits a star.

The inverse square law does not diminish the brightness since the number of stars at a given distance increases as the square of the distance, so the two factors cancel each other out.

The reasons our night sky is not as bright as the surface of the sun is:

The universe is 13.7 +/- 0.15 billion years old so that light from
stars that are further away than 13.7 billion light years could not
have reached us.

The universe is expanding so the light from stars and galaxies that
are far away from us will be redshifted and will not be in the
visible part of the light spectrum.

There is dust and other dark objects that can absorb light from more distant stars.

There is one other reason why the sky should be as bright(er) than a stars surface. Up until 380,000 years after the Big Bang, the entire universe was filled with a very hot plasma. At 380,000 years it had cooled enough for the electrons and protons/nuclei to recombine and produce neutral hydrogen and neutral helium etc. At that time the universe became transparent and was filled with the photons from that the hot ionized plasma. However since then the continued expansion of the universe has redshifted those visible light photons down into the microwave range. That is the source of the Cosmic Microwave Bacground (CMB) radiation that now has an effective temperature of 2.7 degrees Kelvin. However, there are approximately 1,000,000,000 of those photons per proton in the universe so the sky is still quite bright, just not in photons that our eye can see.

Youth and expansion can resolve Olber's Paradox, but dust alone cannot. If indeed we were in equilibrium, the dust would be in thermal equilibrium with the surfaces of stars, and so it would glow with a blackbody spectrum at several thousand Kelvin.
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Chris WhiteAug 24 '12 at 22:11

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That is true once there is enough time for the dust to reach equilibrium. So the dust just extends the youth some extra time...
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FrankHAug 25 '12 at 0:57